Stress spectra of steel highway bridges under traffic loads
L. FRYBA, Institute of Theoretical and Applied Mechanics, Prague, Czechoslovakia
Sresses in three steel highway bridges under usual traffic loads were recoreded and classified
using the rain-flow counting method. The statistical evaluation of stress ranges provided an empirical formula \Jhere the number of heavy vehicles per day and the span of the investigated bridge element are the most important parameters. The formula presents the strees spectra, i.e. the tables of
the number of stress ranges in various stress classes which may be applied to the design of steel
bridges for fatigue.
1. INTRODUCTION
It is well known that the load of highway
bridges according to the national standards
(codes) is much greater than the usual traffic
loads. The reason is the high life of bridges
(about 100 years) in which the real loads may
increase and the necessary safety and reliability.
Therefore, the axle loads of highway vehicles
Here recorded under usual everyday traffic,
(ref. 1) and, later on, also their response on
three steel highway bridges, (ref. 2).
The response of all heavy vehicles was measured during several days, however, the effect of
light personal cars and motorcycles was neglected because they hardly contribute to the stress
ranges.
The number of recorded vehicles is compared
with the number of heavy vehicles according to
the official traffic statistics (year 1980) in
the Table 1.
The stress ranges ~6" due to the standard
loads according to theSCzechoslovak National Code (ref. 5) was also calculated for all important bridge elements.
2. EXPERIMENTS
3. EVALUATION OF EXPERIMENTS
Three steel highHay bridges were chosen for
experiments:
1. Plate girder, span 13.12 m,
2. Box girder with orthotropic roadway, span
45 m,
3. Truss girder, span 44.B m.
The technical description of bridges may be
found in ref. 2.
The stress-time histories were recorded in all
important elements of bridges, i.e. in main girders, cross-girders, stringers, diagonals, etc.
The evaluation of experimental results was automatized using the tape-recorder, osciloscope,
sample analyzer, computer and plotter. The main
results are given in the form of histograms of
numbers of stress ranges an example of which is
reproduced in Fig. 1.
The horizontal a~is in Fig. 1 gives the stress
ranges fl'C. in N/mm (MPa) ~Jhile the vertical
axis represents the number of stress ranges n.
during experiments. The rain-flow counting meihod (ref. 3) was applied for the classification
of stress-ranges into individual stress classes
L::.O .. This method is ~Jell suited for bridges as
can1be shmm for the case of railvJay bridges,
(ref. 4).
For the evaluation the following cases were
distinguished:
- light lorry vehicles up to 5 t of total mass,
- lorry vehicles,
- buses,
- other heavy vehicles,
- all heavy vehicles together.
Table 1. Number of vehicles
Bridge
No.
1
2
3
'-'
152
Number of vehicles
per day,
recorded
from the traffic statistics, 1980
304
387
518
693
119
421
,----,-~-
Heavyvehides and roads: technology, safety and policy. Thomas Telford, London, 1992.
VEHICLE - BRIDGE INTERActION
AO.1 (MPa)
o
15
5
Fig. 1. Stress spectrum,bridge No 2, span 45 m, main girder, 518 vehicles (all together)
As it was not possible to carry out experiments during the whole year (financial reasons)
the results were extrapolated for an annual
traffic in the following way:
- the number of stress cycles was calculated
for an average day,
- the ratio of the number of heavy vehicles
from the official statistics to the number of
that one recorded in an average day was expressed,
- the ratio of the number of heavy vehicles
from the official statistics to the number 500
was calculated because the number of 500 heavy
vehicles per day was taken as a basis,
- the daily stress spectrum was multiplied by
365 and in this way the annual spectrum was obtained.
The described extrapolation resulted in the
annual spectra of stress ranges in all investigated bridge elements ~Jhich ~Jere estimated for
the traffic of 500 heavy vehicles in 24 hours.
4. STATISTICAL EVALUATION OF EXPERIMENTS
The set of experimental data represents the
number of stress cycles n. in stress classes
~er The multidimensional surface fitting giveslthe data that fit experiments with 50% reliability if the method of least squares is applied. However, the limit state design theory
requires the loading data to be obtained with
95% reliability. Therefore, the data obtained
by regression analysis should be enlarged on
1.65 s were s is the standard deviation. Assuming the normal distribution of experimental data around the regression surface the suggested
approach assures after these operations the 95%
reliability of results.
The number ni of stress cycles in the ~th
..
class is affected by the following parameters:
T - average number of heavy vehicles per day,
L - characteristic length of the investigated
bridge element (usually the span),
A- = ~6-.lD.b- - dimensionless stress range
where dO: lis the maximum stress range due to the
standardSmoving load multiplied by the dynamic
impact factor.
As the average daily intensity of heavy vehicles counted according to the statistics around
500 for all investigated bridges it was assumed
the regression surface only for hJO independent
variables Land ;\i:
(l)
The optimum distribution was looked out for a
multidimensional regression that could be transformed by logarithmisation to a linear regression. The exponential, power-exponential, powerpower-exponential, linear-hyperbolic and power
distribution was tested and the last one has
shown as the optimum distribution:
ni
T
= -soo-
a
Lb '\ c.
A 1
(2)
where a, b, c are the regression coefficients.
The assumed regression with the 95% reliability takes the form
0)
where s is the standard deviation, and
k = 1.65 assures the 95% reliability.
153
REAVYVEHICLES AND ROADS
Table 2. Number and amplitude of stress cycles of highway bridges per year
Average
daily
intensity
of heavy
vehicles
Length
L
(m)
Ampli tu de of stress cycles Ai
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Number of stress cycles ni per year (in thousands)
2
3
4
5
10
15
20
30
50
100
500
7
5
5
4
3
2
2
2
1
1
100
900
200
700
400
800
500
000
600
200
190
160
140
130
92
76
67
55
44
32
5.2
4.3
3.8
3.4
2.5
2.1
1.8
1.5
1.2
0.86
23
19
17
15
11
9.2
8.1
6.7
5.3
3.8
The statistical evaluation of all experimental results has provided the following coefficients valid for highway bridges (similar data
for railway bridges may be found in ref. 4):
13.098 79
a
b = - 0.460 76
(4)
c = - 5.208 54
0.929 62
s =
1.65
k =
Ti
Average daily
intensity of
heavy vehicles
T
Tl very heavy
T2 heavy
T3 medium
14 light
3501 to 10 000
1501 to 3 500
501 to 1 500
to
500
154
0.28
0.24
0.21
0.19
0.14
0.11
0.098
0.081
0.064
0.047
0.140.12
0.10
0.092
0.067
0.056
0.049
0.041
0.032
0.023
0.076
0.063
0.056
0.050
0.036
0.030
0.026
0.022
0.017
0.013
0.044
0.037
0.032
0.029
0.021
0.017
0.015
0.013
:
O.OlD
i
0.007
I
5. CLASSIFICATION OF HIGHWAY TRAFFIC
Fatigue of bridges is affected especially by
the heavy vehicles ~Jhile the effect of light
vehicles is negligible. Therefore, the suggested classification in Tab. 3 reflects this fact.
- they include both static and dynamic response of bridges,
- stress spectra include a great number of
ImJ cycles vJhile the great stress ranges are
rare,
Traffic classes
0.63
0.52
0.46
0.41
0.30
0.25
0.22
0.18
0.14
0.10
- they depend on intensity and on composition
of traffic. The direct proportionality with the
average daily intensity of heavy vehicles is
assumed if1 Tab. 2.)
- they depend on the characteristic length of
the investigated bridge element (on the length
and form of the pertinent influence line).
The Tab. 2 was taken over in the Czechoslovak
standard, ref. 5, for loading of bridges. Its
data serve for the calculation of the traffic
load factor \Jhich may be applied to the estimation of fatigue strength and fatigue life of
bridges, (ref. 4).
Using these coefficients the Table 2 VJas calculated. The regression surface is limited by
the class A. = 1 because the static strength
of the element is exhausted for A." 1. The results are rounded by two first valid numbers
and they are given in thousands per year.
The stress spectra have shmm the following
properties:
Table 3. Classification of highway traffic with
1.6
1.4
1.2
1.1
0.78
0.65
0.57
0.47
0.37
0.27
respect to the fatigue of bridges
Length
of nehmrk
%
2
10
37
51
Approximate
average daily
intensity of
all vehicles
30
11
4
1
000
000
500
500
Ratio
Ti /T4
20
7
3
1
VEHICLE - BRIDGE INTERACTION
6. CONCLUSIONS
The experiments carried out on 3 steel high~Jay
bridges presented the stress spectra, i.e. the
tables of number of stress cycles in appropriate
stress classes under usual traffic flow.
The stress spectra include both static and dynamic components of stresses and they depend on
intensity and composition of traffic flow.
A great number of lmJ cycles appears in stress
spectra \vhile the great stress ranges are rare.
It has appeared that only heavy vehicles
should be counted while the effect of light vehicles is negligible with respect to the fatigua
The classification of highway traffic reflects
this fact.
The statistical evaluation of stress ranges
has resulted in an empirical formula for the
calculation of stress cycles in each stress
class where the number of heavy vehicles per day
and the span of the investigated element are the
most important parameters.
The stress spectra may be applied to the design of steel bridges for fatigue and to the estimation of their fatigue life.
REFERENCES
1. Tichy L. Estimation of axle loads using a
highway weighing machine (in Czech). Silnicni
obzor, 1982, vol.43, No.9, 262-266
2. Fryba L. Stress amplitude frequencies in
highway bridges made of steel (in Czech).
Sbornik praci VUZ, 1985, vol.33, 77-111
3. Dowling N.E. Fatigue failure predictions for
complicated stress-strain histories. J.Mater.
1972, vol.7, No.l, 71-87
4. Fryba L. Dynamics of railway bridges. Ellis
Horwood, Chichester - in press
5. Czechoslovak Standard CSN 73 6203 Loading of
bridges (in Czech), 1986
155